To our knowledge, the most general framework in this context has been formalized in [1]}, with the distribution of the error component left unspecified, except for the conditional moments. The motivation for their Generalized Method of Moments (GMM) approach relies on the limitations of complete parametric specifications of the error distribution, that is: (i) parametric distributions restricted to the positive orthant are often not flexible enough or they lack a closed-form density function;
(ii) copulas often imply unrealistic symmetries, do not always model adequately the association among components of the error term, and require expensive tuning in cases of different marginal distributions;
(iii) the exact distribution of the error term may be not interesting, in particular when the analysis is mainly focused on the dynamics of the conditional mean.
Even if [1]} permits a more flexible model specification that is not forced by unhandy parametric assumptions, on the other it introduces problems related to the arbitrary choice of the first two moment conditions. Such a choice can have a crucial impact on the performance on the asymptotic approximation of a GMM estimator, especially when the degree of overidentification is large [3]}. Furthermore, both the recursive model and the general form of vMEM in [1]} face non-negativity challenges, since restrictions to the positive orthant are not explicitely designed [5]}.
